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History of Mathematics

History of Mathematics I INTRODUCTION mathematics, study of relationships between quantities, sizes and properties and of logical operations, with unknown quantities, sizes and properties can be derived. In the past, the mathematics as a science viewed from the crowd, if the sizes as in geometry or the numbers, as in arithmetic, or generalization of these two fields, as in algebra. By the mid-19th Century came more than mathematics the science of relations, or as the science moves considered to be necessary conclusions. This latter view includes mathematical or symbolic logic and Sciences the use of symbols to provide an exact theory of logical deduction and inference to definitions, axioms, postulates and rules for the transformation of primitive elements in complex relationships and theorems. This brief overview of the history of mathematics follows the development of mathematical ideas and concepts, beginning in prehistory. In fact, mathematics is almost as old as humanity itself: evidence for a sense of interest in geometry and geometric pattern has been found in the drafts prehistoric pottery and textiles and cave paintings. Primitive counting systems were almost certainly based on the fingers of one hand or both hands, as demonstrated by the predominance of the numbers 5 and 10 as the basis for most systems today, many of them. Ancient Mathematics II The earliest records of advanced, organized Mathematics come from ancient Mesopotamia, Babylonia and Egypt of the 3rd Millennium BC. It was dominated by arithmetic, mathematics, with an emphasis on the measurement and Calculation of the geometry and follow the later without mathematical concepts such as axioms or proofs. The earliest Egyptian texts, written about 1800 BC, reveal a decimal numbering system, with separate symbols for the successive powers of 10 (1, 10, 100, etc.), as well as in the system used by the Romans. Figures were by writing down the symbol of 1, 10, 100 and so on represented as often as the unit was in a certain number. For example, the letter was written for a five written plays for the representation of the number 5, the symbol for 10 was written six times the number 60 represented, and the symbol for 100 times the number 300 was to represent. Together these symbols represent the number 365th In addition, the total units separated, 10s, 100s and so on have to do in the figures. Multiplication was based on successive Doublings, and division was based on the inverse of this process. The Egyptians used sums of unit fractions (?), Supplemented by the breach?, On all the other groups to express. For example, the break? was the sum of the fractions? and?. With this system, the Egyptians were able to solve all the problems that the arithmetic participating groups, as well as some elementary problems of algebra. In geometry, the Egyptians came at the right rules for finding areas of triangles, rectangles, trapezoids and, and for finding volumes of figures such as bricks, cylinders, and, of course, pyramids. To the area of a circle, the Egyptians used the space? the known diameter of the circle, a value close to the value of the ratio as a PI, but really, at 3.16 instead of PI value of about 3.14. The Babylonian system of numbering was different as the Egyptian system. In the Babylonian system, consisting of clay tablets with different wedge-shaped marks, indicated a single wedge and a wedge-like an arrow stood for 10 (see table). Numbers up to 59 have been formed by these symbols by the additive system, as in Egyptian mathematics. The number 60 was, however, by the same symbol was represented as one, and use that point to a positional symbol. That is, the value of one of the first 59 digits from then hung on to its position in the total number of words. To Sample, followed consisting of a number of a symbol for a 2 for 27 and ends in a stand of 10 for 2 × 602 × 60 + 27 + 10 This principle was extended to the representation of political groups as well, so that the above sequence of numbers could represent just over 2 × 60 + 27 + 10 × (?), or 2 + 27 × (?) + 10 × (? -2). This sexagesimal (base 60), as it is called, the Babylonians had a numerical system as convenient as the decimal (base 10) system. The Babylonians in the time developed a sophisticated mathematics with which they could find the positive roots of a quadratic equation. You could even Searching for the roots of certain cubic equations. The Babylonians had a number of tables, including tables for multiplication and division, tables of squares and tables of compound interest. They could solve problems with complicated Pythagoras, one of its tables contain integer solutions of the Pythagorean equation a2 + b2 = c2, so that c2/a2 decreases steadily from 2 to about arranged?. The Babylonians were also able to sum not only arithmetic and some geometric series, but also sequences of squares. They also came to a good approximation for?. In geometry, they calculated the area of rectangles, triangles, and trapezoids, and the volumes of simple shapes such as stones and bottles. However, the Babylonians did not reach the correct formula for the volume of a pyramid. A Greek Mathematics The Greeks adopted elements of mathematics by both the Babylonians and the Egyptians. The new element in Greek mathematics, however, was the invention of an abstract mathematics founded on a logical structure of definitions, Axioms and proofs. According to later Greek accounts, this development began in the 6th Century BC with Thales of Miletus and Pythagoras of Samos, the latter a religious leader, study of the significance of the numbers to understand the world taught. Some of his students made important discoveries about the theory of numbers and geometry, which attributed all were Pythagoras. In the 5th Century BC, were some of the great geometer of the atomistic philosopher Democritus of Abdera, who discovered the right formula for the volume of a pyramid, and Hippocrates of Kos, which limits the areas of the crescent-shaped pieces of circular arcs are the same areas of certain triangles discovered. This discovery is the famous problem squaring the circle in the context is that the construction of a square area equal in a given circle. Two other famous mathematical problems while were of the century the trisection of an angle and doubling a cube, that is, the construction of a cube whose volume is twice that of a certain cube. All these problems have been solved, and in a variety of ways, all of which complicate the use of instruments as a straight edge and a geometric compass. Only in the 19th Century has shown that the three above-mentioned problems has never been solved using these tools alone. In the second half of the 5th Century BC, discovered an unknown mathematician, that no unit of length, both the side and diagonal of a square to measure. That is, the two lengths are incommensurable. This means that no counting numbers n and m are, the ratio expresses the ratio of the page on the diagonal. Since the Greeks than just the numbers (1, 2, 3 and so on) as numbers, they had no access to such numerical ratio of the diagonal to rule on the side. (This ratio?, Dignity is as irrational now.) As a consequence of the Pythagorean theory of the relationship, based on figures, to be abandoned and a new, imported non-numerical theory. This was the fourth in Century BC mathematician Eudoxus of Cnidus, whose solution can be found the elements of Euclid did. Eudoxus also discovered a method for rigorously prove statements about land and volumes by successive approximations. Euclid was a mathematician and teacher who worked at the famous Museum of Alexandria and also wrote on optics, astronomy and music. The 13 books, from which its elements contain much of the basic mathematical skills by the end of the 4th Century BC to the geometry of polygons and discovered the Circle, number theory, the theory of incommensurables, solid geometry and the elementary theory of areas and volumes. The century that followed was marked by Euclid mathematical Brilliance, as displayed in the works of Archimedes of Syracuse and a younger contemporary, Apollonius of Perga. Archimedes method used in the discovery, in theory, on the basis weighing infinitely thin slices of numbers to find the areas and volumes of figures from the conic sections. These conic sections had been discovered by a pupil of Eudoxus Menaechmus named, and they were the subject of a treatise by Euclid, Archimedes, but 'writings on them are the earliest to survive. Archimedes studied the priorities and the stability of the different solids floating in water. Many of his works is part of the tradition, led in the 17th Century, to the discovery of calculus. Archimedes was a Roman soldier during the sack of Syracuse killed. His younger contemporary, Apollonius, produces an eight-book treatise on conic sections, that the names of the sections: ellipse, parabola and hyperbola established. Moreover, the basic treatment offered by their geometry by the time the French philosopher and scientist René Descartes in the 17th Century. According to Euclid, Archimedes and Apollonius, Greece produced no comparable stature surveyor. The writings of Heron of Alexandria in the first Century AD show how elements from the Babylonian and Egyptian traditions survived mensurational arithmetic in addition to the logical structures of the great geometer. Very much in the same Tradition, but with more difficult problems are the books of Diophantus of Alexandria in the third Century AD. They deal with the search for rational solutions, two Types of problems, directly to equations in several unknowns. Such equations are now called Diophantine equations and are the subject of Diophantine analysis. B Applied Mathematics in Greece described in parallel with the studies in pure mathematics studies in optics, mechanics, astronomy, and were. Many of the greatest mathematical writer like Euclid and Archimedes, also wrote on astronomical topics. Shortly after the time of Apollonius, Greek astronomers adopted the Babylonian system of collecting and political groups, for example to same time composing tables of chords in a circle. For a circle of some fixed radius, such as tables give the length of the chords subtending a sequence of arcs Increasing by some fixed amount. They are synonymous with a modern sine table, and their composition marked the beginnings of trigonometry. In the earliest such tables, which increased Hipparchus in about 150 BC, the sheets in increments of 7? °, 0 ° to 180 °. At the time of the astronomer Ptolemy in the 2nd Century AD Greek Championship the numerical procedure was to the point where Ptolemy in his Almagest was a table of chords in a circle are advanced for the steps? °,, although they expressed sexagesimally, just to about five decimal places. In the meantime, methods were developed to solve problems with flat triangles, and a set-named after the astronomer Menelaus of Alexandria, was to search for specific lengths of arc on a sphere, when other forms are known established. This progress was Greek astronomers, which they solve the problems of spherical astronomy and to develop an astronomical system that prevailed until the time of the German astronomer Johannes Kepler needed. III of the Middle Ages and the Renaissance mathematics from the time of Ptolemy, a tradition of research into the mathematical masterpieces of the last century in various centers of Greek learning was established. The preservation of such works as have survived into modern times began with this tradition. It is in the Islamic world, where the original developments based on these masterpieces was continued appeared the first time. The earliest developments based on these original masterpieces, but not in such centers tradition appear, but in the Islamic world. An Islamic and Indian mathematics After a century of expansion, in which the religion of Islam spread from its beginnings in the Arabian peninsula to an area dominated from Spain to the borders of China, started the Muslims, the results of the acquisition of "foreign sciences ". In support centers such as the House of Wisdom in Baghdad by the ruling caliphs and wealthy individuals, produced Arabic versions of Greek Translator and Indian mathematical works. With the year completed the acquisition was 900, and Muslim scholars began to build what they had acquired. So extended, the Hindu mathematician decimal positional system of arithmetic of whole number, decimals, and the 12th Century Persian mathematician Omar Khayyam Hindu generalized methods for extracting Square and cube roots belong fourth, fifth and higher roots. In algebra, closed al-Muhammad al-Khwarizmi Karaji the algebra of polynomials are polynomials even with an infinite number of terms. (Al-Khwarizmi name, by the way, is the source of the word algorithm, and the title of one of his books is the source of the word algebra.) Ibrahim ibn Sinan continue as geometer Archimedes' investigations of surfaces and volumes, and Kamal al-Din and other application of the theory of conics to optical problems to solve. The Hindu sine function and Menelaus' theorem, mathematicians from Habas al-Hasib to Nasir ad-Din at-Tusi created the mathematical disciplines of plane and spherical trigonometry. This non-mathematical disciplines in the West only since the publication of De Triangulis Omnimodibus the German astronomer Regiomontanus. Finally, a number of Muslim Mathematicians made important discoveries in number theory, while others, a variety of numerical methods for solving said equations. The Latin West acquired much of this learning in the 12th Century, the great century of translation. Along with translations of Greek classics, these works were responsible for Muslim the growth of mathematics in the West during the late Middle Ages. Italian mathematician Leonardo Fibonacci, and as Luca Pacioli, one of the many writers of the 15th Century on algebra and arithmetic for merchants, depended heavily on Arabic sources for their knowledge. IV Western Renaissance MATHEMATICS Although the late Middle Ages saw some fruitful mathematical considerations of problems of infinity of writers such as Nicole Oresme, it was not until the early 16th Century that a truly important mathematical discovery was made in the West. The discovery of an algebraic formula for the solution of both the cubic and biquadratic equations, was 1545 by the Italian mathematician Geronimo Cardano in his Ars Magna published. The discovery attracted the attention of mathematicians to complex numbers and stimulated a search for the solutions of equations of degree higher than four. It was this search, in turn, that the initial work on group theory led at the end of the 18th Century, and the French mathematician Galois Évariste 'Theory of equations in the early 19 Century. The 16th Century also saw the beginnings of modern algebraic and mathematical symbols and the remarkable work on the solution of equations by the French mathematician François Viète. His writings influenced many mathematicians of the next century, including Pierre de Fermat in France and Isaac Newton in England. V MATHEMATICAL Since 16 Century, Europeans dominated in the development of mathematics after the Renaissance. A 17 In the 17th century Century were the most progress made in mathematics since the time of Archimedes and Apollonius. The century began with the discovery by the Scottish mathematician John Napier of logarithms, the maintenance of utility asked to note the French astronomer Pierre Simon Laplace, almost two centuries later that Napier, by the Halving the work of astronomers doubled, had in their lifetime. The science of number theory, which had lain dormant since the Middle Ages, illustrated from the 17th Century Advances built on ancient learning. It was suggested Diophantus' Arithmetica that Fermat significantly to the theory of numbers ahead. His most important assumption in the field to the margin of his copy of Arithmetica was written, that no solutions to the + bn = cn exist for positive integers a, b and c, if n is greater than 2. This Conjecture, known as Fermat's Last Theorem, stimulates very important work in algebra and number theory, only in 1995, the sentence proved satisfactory, with the help of Andrew Wiles by Richard Taylor. Two important developments in pure geometry occurred during the century. The first one was the publication, in Discourse on Method (1637) by Descartes, of his discovery of analytic geometry, which showed how to do the algebra, which had developed since the Renaissance used to study the geometry of curves. (Fermat made the same discovery but do not publish.) This book, along with short essays, which had been published with it, excited, and the basis for the mathematical Isaac Newton's work in the 1660s. The second development in geometry was the publication by the French engineer Gérard Desargues in 1639 of his discovery projective geometry. Although the work was far from Descartes and the French philosopher and scientist Blaise Pascal, appreciate his eccentric terminology and the excitement the previous publication of the analytical geometry retarded the development of their ideas to the beginning of the 19th Century and the works of the French mathematician Jean Victor Poncelet. Another important step in mathematics in the 17th Century was the beginning of probability theory in the correspondence of Pascal and Fermat to a problem in the game, as the Problem of points. These unpublished stimulates the Dutch scientist Christiaan Huygens to publish work on a small treatise on probabilities in dice Games, which was to guess by the Swiss mathematician Jakob Bernoulli in his art reprinted. Both Bernoulli and French mathematician Abraham de Moivre, in his doctrine of chances in 1718, the newly discovered calculus applied to rapid progress in the theory, which had until then to make important applications in the rapidly developing insurance industry. Without question, however, was the crowning event of the 17 mathematical Century, Newton's discovery, 1664-1666, the differential and integral calculus. With this discovery, Newton on earlier work by his colleagues Englishman John Wallis and Isaac Barrow, and on the work of mathematicians such as Continental, as Descartes built, Francesco Bonaventura Cavalieri, Johann Hudde and Gilles Personne de Roberval. About eight years later, as Newton, the not yet published his discovery, the German Gottfried Wilhelm Leibniz calculus rediscovered and published first in 1684 and 1686th Leibniz notation systems, such as dx, are today in the calculus. B 18 The rest of the 17th century Century and a good part of the 18th have been through the work of students from Newton and Leibniz, the ideas used to solve a variety of problems in physics, astronomy made and engineering. In the course of doing so they also created new areas of mathematics. For example, Johann and Jakob Bernoulli invented the calculus of variations and French mathematician Gaspard Monge invented differential geometry. In France, Joseph Louis Lagrange was a purely analytical treatment of the mechanics in his big Analytical Mechanics (1788), in which he declared The famous Lagrange equations for a dynamic system. He contributed to differential equations and number theory, but also, and led to the theory of groups. His contemporary, Laplace, wrote the Analytical Theory of Probabilities (1812) and the classic Celestial Mechanics (1799-1825), earned him the title of the "French Newton". The largest Mathematicians of the 18th Century was Leonhard Euler, a Swiss, who made fundamental contributions to the calculus and to all other branches of mathematics, and applications of mathematics. He wrote textbooks on calculus, algebra and mechanics, the models were for writing styles in these areas. The success of Euler and other mathematicians to solve with calculus in the mathematical and physical problems, but only accentuates their failure to develop a satisfactory explanation of its basic ideas. That is, Newton have their own accounts on kinematics and velocities, Leibniz statement was based on infinitesimals and Lagrange's treatment was based purely algebraic and founded on the idea of infinite series. All these systems were unsatisfactory when measured against the logical standards of Greek geometry, and the problem was not solved until the following century. C 19 Century In 1821 a French mathematician Augustin Louis Cauchy type, in a logically satisfactory approach managed to calculus. He based its decision only on finite quantities, and the idea of limits. This solution was a further problem, however, that the logical definition of the term "real number". Although the statement of Cauchy calculus based on this idea, it was not Cauchy but the German mathematician, Julius WR Dedekind, of a satisfactory definition of real numbers found in relation to the rational numbers. This definition is still being taught, but also other definitions have been endorsed by both the German mathematician Georg Cantor and Karl Weierstrass TW given. Another important issue that arose from the problem, first stated in the 18 Century describe the motion of a vibrating string, was the definition of what determined by the function. Euler, Lagrange, and French mathematician Jean-Baptiste Fourier, all contributed to the solution, but it was the German mathematician Peter GL Dirichlet, the definition in terms of a correspondence between the elements of the domain and the area proposed. This is the definition that is found in texts today. In addition, consolidates the foundations of the analysis, as the techniques of the calculus were then called, mathematician 19th Century, great progress made in the subject line. At the beginning of the century, Carl Friedrich Gauss gave a satisfactory explanation of complex numbers, and these numbers then formed a new field for the analysis, one that was developed in the works by Cauchy, Weierstrass and the German mathematician Georg FB Riemann. Another important advance in the Fourier analysis, the study of infinite sums whose terms of trigonometric Functions. Today known as a Fourier series, they are still powerful tools in pure and applied mathematics. In addition, the investigation, which functions could be equal Fourier series led Cantor to the study of infinite sets and to an arithmetic of infinite numbers. Cantor's theory as very abstract and even as a "disease, mathematics will soon recover, "was attacked, now part of the foundations of mathematics and has in more recent applications in study of turbulent flow Fluids. Another 19th Century discovery, which was obviously an abstract and useless at the time was non-Euclidean geometry. In the non-Euclidean geometry, can more than one parallel to a given line through a given point not on the line. Apparently this was discovered first by Gauss, Gauss was terrible, but the controversy that could result from the publication. The same results have been independently rediscovered and published by the Russian mathematician Nikolai Ivanovich Lobachevsky and the Hungarian János Bolyai. Non-Euclidean geometries have been studied in a very general setting of Riemann manifolds with his invention, and since the Work of Einstein in the 20th Century, they have also found applications in physics. Gauss was one of the greatest mathematicians who ever lived. From his diaries Youth show that this prodigy had already made important discoveries in number theory, an area in which his book Disquisitiones Arithmeticae (1801) marks the beginning of modern times. While only 18, Gauss discovered that, a regular polygon with m sides are constructed and by Straight-Edge compass, if m is a power of two different times Primes of the form 2 n + 1 In his dissertation, he gave the first satisfactory proof of the fundamental theorem of algebra. He often has scientific and mathematical investigations combined. Examples include the development of statistical methods with its investigations into the orbit of a newly discovered planetoid, his foundation work on the field of potential theory, together with the study of magnetism, and his study of the geometry of the curved surfaces in combination with its studies on the Surveying. Of greater significance for algebra itself as Gauss's proof of his fundamental theorem, the transformation of the subject was in the 19 Century from a study of polynomials, a study on the structure of algebraic systems. An important step in this direction was the invention of symbolic algebra in England by George Peacock. Another was the discovery of algebraic systems, many but not all, of the properties of real numbers. Such systems are the quaternions of the Irish mathematician William Rowan Hamilton, the vector analysis of American mathematician and physicist J. Willard Gibbs and the proper n-dimensional spaces of the German mathematician Hermann Günther Grassmann. A third important step was the development of the theory group, from its beginnings in the work of Lagrange. Galois applied this work can be quite a theory of polynomials, where by an algebraic formula can be solved. Just as Descartes had the ring of his time to the study of applied geometry, the German mathematician Felix Klein and the Norwegian Mathematician Marius Sophus Lie algebra of the 19 applied Century. Klein, she applied for the classification of geometries in terms of their groups of transformations (the so-called Erlanger Program), and Lie, they turned on a geometric theory of differential equations known by means of continuous groups of transformations as Lie groups. In the 20th Century Algebra was also a general form of geometry known as topology, applied. Another issue that in the 19 Century was transformed, especially by the English mathematician George Boole's Laws of Thought (1854) and Cantor's set theory was the foundation of mathematics. Towards the end of the century, however, a number of paradoxes in Cantor's theory discovered. Such a paradox, the English mathematician Bertrand Russell, the concept of a series aimed found. Mathematicians responded by theories set sufficiently restrictive to hold the paradoxes from arising, but they left open the question of whether other paradoxes in these theories could be limited, that show whether the theories were consistent. given as of today, but relative consistency proofs have been, that is, theory A is consistent if theory B is consistent.) Particularly disturbing is the result, in 1931 by the American logician Kurt Gödel proved that in any axiom system sophisticated enough to be interesting, most mathematicians, it is possible, sentences, are not the truth held within the system frame. VI CURRENT mathematics at the International Conference of Mathematicians in 1900 held in Paris, said the German mathematician David Hilbert to the Assembly. Hilbert was a professor at the University of Göttingen, the former academic home of Gauss and Riemann. He had to most areas of mathematics has contributed, from his classic Foundations of Geometry (1899), jointly authored methods of mathematical physics. Hilbert's address at the University of Göttingen was a survey of 23 mathematical problems, which he felt would guide the work is in mathematics in the coming century. These problems have encouraged, in fact, much of the mathematical Research of the century. solved as the news breaks that another of the "Hilbert Problems" was, mathematicians in the world are waiting for the details of the story with impatience. More important than these problems, an event that Hilbert was not predictable appears to be intended for an even greater role in future development of math play, namely the invention of the programmable digital computer. Although the roots of the computer back to the calculating machines of Pascal aligned and go Leibniz 17th Century, it was Charles Babbage in the 19th Century England, a machine that could automatically perform calculations on a program of instructions on cards or tape Designed saved. Babbage's imagination overtook the technology of his time, and not until the invention of the relay, then the vacuum tube, and then the transistor, that large Scale, programmed calculation was possible. This development has given great impetus to areas of mathematics, such as numerical analysis and finite mathematics. It has new areas for the mathematical study of how the study of algorithms proposed. It has also become a powerful tool in fields as diverse as number theory, Differential equations and abstract algebra. Moreover, the computer enables the solution of several long-standing problems in mathematics, and as a four-color problem for the first time in the middle of the 19th Century proposed. The theorem states that four colors to color all the cards are, because no two countries with a contiguous Border require different colors. The sentence was finally in 1976 by a large-scale computer proved at the University of Illinois. Mathematical knowledge in the modern World is at a faster rate than ever before advancing. Theories that were once separate are, in both theories, comprehensive and abstract have been accepted. Although many important issues have been resolved, others remain to be perennial, as the Riemann hypothesis, and new, equally challenging problems arise. Even the abstract Mathematics seems to be finding applications. About the Author

PRABHAT MARWAHA
M.SC MATHS, B.ED.
15 YEARS TEACHING EXPERIENCE TO SENIOR CLASSES.
PRESENTLY WORKING AS VICE PRINCIPAL IN JNV
LONGOWAL, SANGRUR, PUNJAB (INDIA).
EMAIL-ID:prabhat.marwaha@gmail.com

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